\(\tilde{h}_{N, T}\)  converges pointwise to \(\tilde{h}(y)= {(y^{\gamma }-1)}/{\gamma }\)  as \(N, T\rightarrow \infty \)  by theorem REF  and eq. (REF ) with \(U_T\)  in place of \(U_F\) .
Since the limit is continuous, and \(\tilde{h}_{N, T}(y)\)  is monotonic in \(y\) , convergence is locally uniform in \(y\)  (see section 0.1 in [1]}). Define \(y_N=N(1-\exp ({-E_1^*/N}))\) . Then \(y_N\rightarrow E_1^*\)  and \(y_N\)  is a bounded sequence. Then as \(N, T\rightarrow \infty \)  it holds that \(\tilde{h}_{N, T}(y_N^{-1})\rightarrow \tilde{h}((E_1^*)^{-1})\)  (observe that \(E_1^*\)  does not depend on \(N\)  or \(T\) ).
